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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 12th 2013
   • (edited Sep 12th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRecall (or else notice) that one of the characteristics of the "[[holographic principle]]" in QFT is that it relates fields of a bulk field theory to _sources_ of a boundary field theory. Thanks to some hints from Igor Khavkine, I realized that there is a formalization of this statement for the case that the bulk theory is the 2d Chern-Simons theory which is the non-perturbative Poisson sigma model, whose boundary theory is the history space of a physical system with its "[[off-shell Poisson bracket]]". In this case the moduli stack of bulk fields is precisely the space of sources of the boundary theory. This follows by just combining on the one hand the discussion at _[[off-shell Poisson bracket]]_, which is in principle "well known" (see the detailed citations there for what is "known", I would instead say that Igor Khavkine was the one who fully realized this, expressed to me in private communication), with, on the other hand, the description of Poisson manifolds as boundaries of the non-perturbative Poisson sigma-model as Joost Nuiten described in [[schreiber:master thesis Nuiten|his thesis]] (see also the Examples-section at [[motivic quantization]]). I have added a remark explaining this "Poisson holography"-statement at _[off-shell Poisson bracket -- boundary theory field interpretation](http://ncatlab.org/nlab/show/off-shell+Poisson+bracket#BoundaryFieldTheoryInterpretation)_.

   Recall (or else notice) that one of the characteristics of the “holographic principle” in QFT is that it relates fields of a bulk field theory to sources of a boundary field theory.

   Thanks to some hints from Igor Khavkine, I realized that there is a formalization of this statement for the case that the bulk theory is the 2d Chern-Simons theory which is the non-perturbative Poisson sigma model, whose boundary theory is the history space of a physical system with its “off-shell Poisson bracket”. In this case the moduli stack of bulk fields is precisely the space of sources of the boundary theory.

   This follows by just combining on the one hand the discussion at off-shell Poisson bracket, which is in principle “well known” (see the detailed citations there for what is “known”, I would instead say that Igor Khavkine was the one who fully realized this, expressed to me in private communication), with, on the other hand, the description of Poisson manifolds as boundaries of the non-perturbative Poisson sigma-model as Joost Nuiten described in his thesis (see also the Examples-section at motivic quantization).

   I have added a remark explaining this “Poisson holography”-statement at off-shell Poisson bracket – boundary theory field interpretation.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 10th 2023
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   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Eugene Rabinovich]], *A Classical Bulk-Boundary Correspondence* &lbrack;[arXiv:2202.12332](https://arxiv.org/abs/2202.12332)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Poisson+holography/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poisson+holography/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Poisson+holography">current</a>

   added pointer to:

   • Eugene Rabinovich, A Classical Bulk-Boundary Correspondence [arXiv:2202.12332]

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness, I added pointer also to: * [[Joost Nuiten]], Sec. 5.2.2 in: *[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]* (August 2013) <a href="https://ncatlab.org/nlab/revision/diff/Poisson+holography/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poisson+holography/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Poisson+holography">current</a>

   for completeness, I added pointer also to:

   • Joost Nuiten, Sec. 5.2.2 in: Cohomological quantization of local prequantum boundary field theory (August 2013)

   diff, v4, current